Meshless Methods Introduction - Meshless Methods in Biomechanics - page 44

Fig. 3.3 a Influence-domains with different sizes and shapes. b Example of a bad choice in the
size of the influence-domain, interest point x 1 has much more nodes inside the influence-domain
when compared with interest point x 2
and u i ð x I Þ is the approximation or interpolation function value of the ith node
obtained using only the n nodes inside the influence-domain.
Establishing the equation system is the next step. In meshless methods, the
discrete equations can be formulated using the approximation or interpolation
functions applied to the strong or weak form formulation. In the case of the
meshless methods using the weak form of Galerkin, the discrete equations can be
obtained applying to the differential equation governing the physic phenomenon
the weighted residual method of Galerkin. The produced equations are then
arranged in a local nodal matrix form and assembled into a global equation system
matrix. In the case of a static problem the global equation system matrix is a set of
algebraic equations, for the case of the free vibration analysis or buckling analysis
is a set of eigenvalue equations and in the case of a dynamic (time dependent)
analysis is a set of differential equations.
In order to obtain the distinct solutions for the distinct types of analyses, one
must choose the appropriate solver. For static problems with the global equation
system matrix the displacement field is obtained. To obtain the solution field, a
linear algebraic equation solver is required. In this work for small systems it is
used a Gauss elimination method and for larger equation systems a LU decom-
position method. For the free-vibration and buckling analysis it is required the use